# American Institute of Mathematical Sciences

December  2017, 22(10): 3891-3901. doi: 10.3934/dcdsb.2017200

## A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance

 Department of Mathematics, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan

Received  April 2014 Revised  July 2017 Published  August 2017

The initial value problem for a reaction-diffusion system with discontinuous nonlinearities proposed by Hofbauer in 1999 as an equilibrium selection model in game theory is studied from the viewpoint of the existence and stability of solutions. An equilibrium selection result using the stability of a constant stationary solution is obtained for finite symmetric 2 person games with a 1/2-dominant equilibrium.

Citation: Hideo Deguchi. A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3891-3901. doi: 10.3934/dcdsb.2017200
##### References:
 [1] H. Deguchi, Existence, uniqueness and non-uniqueness of weak solutions of parabolic initial-value problems with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1139-1167. doi: 10.1017/S0308210500004315. [2] H. Deguchi, On weak solutions of parabolic initial value problems with discontinuous nonlinearities, Nonlinear Anal., 63 (2005), e1107-e1117. doi: 10.1016/j.na.2004.10.004. [3] H. Deguchi, Existence, uniqueness and stability of weak solutions of parabolic systems with discontinuous nonlinearities, Monatsh. Math., 156 (2009), 211-231. doi: 10.1007/s00605-008-0082-y. [4] H. Deguchi, Weak solutions of a parabolic system with a discontinuous nonlinearity, Nonlinear Anal., 71 (2009), e2902-e2911. doi: 10.1016/j.na.2009.07.005. [5] I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867. doi: 10.2307/2938230. [6] J. C. Harsanyi and R. Selten, A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge, MA, 1988. [7] J. Hofbauer, Stability for the best response dynamics, preprint, 1995. [8] J. Hofbauer, Equilibrium selection via travelling waves, in Game Theory, Experience, Rationality (eds. W. Leinfellner and E. Köhler), Kluwer Academic Publishers, Dordrecht, 5 (1998), 245-259. [9] J. Hofbauer, The spatially dominant equilibrium of a game, Ann. Oper. Res., 89 (1999), 233-251. doi: 10.1023/A:1018979708014. [10] J. Hofbauer and P. L. Simon, An existence theorem for parabolic equations on $\textbf{R}^N$ with discontinuous nonlinearity, Electron. J. Qual. Theory Differ. Equ., 8 (2001), 1-9. [11] H. P. McKean and V. Moll, Stabilization to the standing wave in a simple caricature of the nerve equation, Comm. Pure Appl. Math., 39 (1986), 485-529. doi: 10.1002/cpa.3160390403. [12] S. Morris, R. Rob and H. S. Shin, p-dominance and belief potential, Econometrica, 63 (1995), 145-157. doi: 10.2307/2951700. [13] D. Oyama, S. Takahashi and J. Hofbauer, Monotone Methods for Equilibrium Selection Under Perfect Foresight Dynamics, Working paper No. 0318, University of Vienna, 2003. doi: 10.2139/ssrn.779864. [14] S. Takahashi, Perfect foresight dynamics in games with linear incentives and time symmetry, Internat. J. Game Theory, 37 (2008), 15-38. doi: 10.1007/s00182-007-0101-6. [15] D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation, SIAM J. Math. Anal., 14 (1983), 1107-1129. doi: 10.1137/0514086.

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##### References:
 [1] H. Deguchi, Existence, uniqueness and non-uniqueness of weak solutions of parabolic initial-value problems with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1139-1167. doi: 10.1017/S0308210500004315. [2] H. Deguchi, On weak solutions of parabolic initial value problems with discontinuous nonlinearities, Nonlinear Anal., 63 (2005), e1107-e1117. doi: 10.1016/j.na.2004.10.004. [3] H. Deguchi, Existence, uniqueness and stability of weak solutions of parabolic systems with discontinuous nonlinearities, Monatsh. Math., 156 (2009), 211-231. doi: 10.1007/s00605-008-0082-y. [4] H. Deguchi, Weak solutions of a parabolic system with a discontinuous nonlinearity, Nonlinear Anal., 71 (2009), e2902-e2911. doi: 10.1016/j.na.2009.07.005. [5] I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867. doi: 10.2307/2938230. [6] J. C. Harsanyi and R. Selten, A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge, MA, 1988. [7] J. Hofbauer, Stability for the best response dynamics, preprint, 1995. [8] J. Hofbauer, Equilibrium selection via travelling waves, in Game Theory, Experience, Rationality (eds. W. Leinfellner and E. Köhler), Kluwer Academic Publishers, Dordrecht, 5 (1998), 245-259. [9] J. Hofbauer, The spatially dominant equilibrium of a game, Ann. Oper. Res., 89 (1999), 233-251. doi: 10.1023/A:1018979708014. [10] J. Hofbauer and P. L. Simon, An existence theorem for parabolic equations on $\textbf{R}^N$ with discontinuous nonlinearity, Electron. J. Qual. Theory Differ. Equ., 8 (2001), 1-9. [11] H. P. McKean and V. Moll, Stabilization to the standing wave in a simple caricature of the nerve equation, Comm. Pure Appl. Math., 39 (1986), 485-529. doi: 10.1002/cpa.3160390403. [12] S. Morris, R. Rob and H. S. Shin, p-dominance and belief potential, Econometrica, 63 (1995), 145-157. doi: 10.2307/2951700. [13] D. Oyama, S. Takahashi and J. Hofbauer, Monotone Methods for Equilibrium Selection Under Perfect Foresight Dynamics, Working paper No. 0318, University of Vienna, 2003. doi: 10.2139/ssrn.779864. [14] S. Takahashi, Perfect foresight dynamics in games with linear incentives and time symmetry, Internat. J. Game Theory, 37 (2008), 15-38. doi: 10.1007/s00182-007-0101-6. [15] D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation, SIAM J. Math. Anal., 14 (1983), 1107-1129. doi: 10.1137/0514086.
Condition (B2)
is the set of ${\bf u} \in \Delta$ to which the pure strategy $i$ (i.e., ${\bf e}_i$) is the best reply in the game $(13)$ for $i = 1, 2, 3$. The three dots denote the middle points of the edges.
is the set of ${\bf u} \in \Delta$ to which the pure strategy $i$ (i.e., ${\bf e}_i$) is the best reply in the game $(14)$ for $i = 1, 2, 3$. The three dots denote the middle points of the edges.
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